A theory that describes how matter produces and responds to the geometry of space and time. It was first published by Einstein in 1915 and is currently used to study the structure and evolution of the universe, as well as having practical applications like GPS.

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How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative?

The Question How does one prove that Rindler's definition of the covariant derivative of a covariant vector field $\lambda_a$ as \begin{align} \lambda_{a;c} = \lambda_{a,c} - \Gamma^{b}_{\ \ ca} ...
3
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1answer
87 views

Interpreting perturbation theory in general relativity

In quantum mechanics we start with a Hamiltonian $H_0$ for which we know the exact eigenstates and energy eigenvalues. We perturb it by a known term $H$, and then attempt to compute (approximately) ...
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1answer
55 views

Why we can set variations for the metric and its derivatives to zero at infinity?

This question is the continuation of the following one. I still don't understand why $(1)$ may be set to zero. This refers to the zero value variations of metric and its derivatives on the infinitely ...
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0answers
57 views

Induced metric on the boundary of a manifold

The Gibbons-Hawking-York term which supplements the Einstein-Hilbert action is, $$S_{GH} = \frac{1}{8\pi G} \int_{\partial M} d^3 x\sqrt{-h} \, K$$ where $\partial M$ is the boundary of the manifold ...
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4answers
426 views

Is the concept of tensor rank useful in physics?

The term 'tensor rank' is sporadically used in the mathematical literature to denote the minimum number of simple terms (i.e. tensor products of vectors) needed to express the tensor. This is ...
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1answer
109 views

Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By ...
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2answers
769 views

Neutrino unaffected by gravity

Are neutrinos affected by gravity? If not, could that be a plausible reason for a neutrino taking a shorter path than light, since light is affected by gravity?
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0answers
36 views

How to calculate the minimum number of extrinsic dimensions of a metric tensor?

The Question How does one calculate the minimum number of dimensions of an extrinsic space that can be used to define the metric tensor \begin{align} g_{mn} = \dfrac{\partial y^k}{\partial x^m} ...
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1answer
164 views

A few questions related to frame dragging

I am trying to get my head around a few concepts related to frame dragging and related physics. In regards to black holes that have no charge and all their mass is tied up in rotational kinetic ...
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39 views

Positive Mass Theorem [duplicate]

I'm a third year maths undergrad doing a project on minimal surfaces. However I'm really struggling to understand what the PMT is trying to explain? Could anyone help explain this (as simply as ...
20
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5answers
5k views

How does gravitational lensing account for Einstein's Cross?

Einstein's Cross has been attributed to gravitational lensing. However, most examples of gravitational lensing are crescents known as Einstein's rings. I can easily understand the rings and crescents, ...
12
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5answers
679 views

The definition of an inertial reference frame in Einstein's relativity

I'm reading Sean Carroll's book on general relativity, and I have a question about the definition of an inertial reference frame. In the first chapter that's dedicated to special relativity, the ...
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1answer
107 views

General relativity, gravity and spacetime curvature [duplicate]

There is a very fundamental flaw in the common explanation given of the space-time curvature due to massive objects. It is said that a massive object curves space time just like a bowling ball on a ...
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0answers
28 views

gravitational lensing [duplicate]

I had read somewhere that a star, whose light passes very close to the sun and reaches the earth produces 4 images of the same star (left, right, top and bottom) in a telescope due to gravitational ...
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0answers
56 views

Reissner-Nordström Black Holes

The Reissner-Nordström black holes are described by the metric, \begin{align} ds^2 = -\left(1-\frac{2M}{r}+\frac{Q^2}{r^2}\right)dt^2 + \frac{1}{1-\frac{2M}{r}+\frac{Q^2}{r^2}}+r^2d\Omega^2 ...
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0answers
37 views

Gravitational redshift of temperature and electrostatic potential

Consider a charged black hole in four-dimensional Minkowski spacetime, with charge $Q$, mass $M>Q$: $ds^2=-f(r)dt^2+\frac{1}{f(r)}dr^2+r^2d\Omega_2^2$, with $f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$. ...
2
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2answers
168 views

How long does it take for a black hole to form?

The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the ...
0
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2answers
166 views

Why does gravity attract non-metallic objects?

Why does gravity attract non-metallic objects as magnetism does? I understand why gravity, because of mass of an object, works. But earth has a magnetic field, and the moon does not. Indeed, many ...
2
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3answers
125 views

Really nothing special when falling into a black hole?

It has been said time and again, that an observer who falls into a black hole will not notice anything special. Is this really true? There is of course the problem with the tidal forces, but I ...
5
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2answers
85 views

Speed of gravity in cosmological codes and ephemeris generation

There are few questions in Phys.SE concerning the speed of gravity, and the answers are traditionally that the speed of gravity equals to the speed of light. But in that case I have three more ...
2
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1answer
91 views

Can Information Travel Faster Than The Speed Of Light? [duplicate]

Many believe that nothing can travel faster than speed of light, not even information. Personally, i think theoretically information can. Consider this following imaginary experiment: Imagine we are ...
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1answer
38 views

Smaller mass in gravity well?

When sitting in a gravity well, as we do on earth, does our effective mass become smaller than our rest mass due to having negative potential energy? Correspondingly, does a free falling mass (from ...
4
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2answers
276 views

Can special/general relativity be derived from the standard model?

Can special/general relativity be derived from the standard model? For example the time dilatation in strong gravitation? My feeling is yes, but I am not quite sure.
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0answers
26 views

Physical difference between the derivative of the Hubble parameter and the second derivative of the scale factor?

In Carroll's GR book, he discusses the difference between $\dot H$ and $\ddot a$, stating that they are the answers to two different questions, pg. 349. He seems to imply that if one set up two ...
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0answers
107 views

How explain this perturbing equation about the 43 arcseconds?

The planetary orbits have been studied as ellipses but the solar system is in motion in relation to the distant stars. Their path is along the tip of an helix and the ecliptic plane is a convenient ...
2
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3answers
181 views

D'Alembertian for a scalar field

I have read that the D'Alembertian for a scalar field is $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu). $$ Exactly when is this correct? Only for ...
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0answers
24 views

How is the scale factor from the FLRW equation used with Volume?

I'm trying to put a spreadsheet together that shows the co-moving volume of the universe from the time soon after the Big Bang through the present and then as predicted into the future. I am pretty ...
3
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1answer
116 views

Why do galaxies “dissappear?”

So, this is a dumb question but a bit of information confused me lately. Before, I figured galaxies were no longer visible by us because their luminosity decreased in an inverse square manner. ...
2
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5answers
405 views

Is there a universal rest frame of reference?

I am still struggling with C being a constant and what that implies. So can an experiment be done to find the resting state for the universe? Take a device with an observer and a light source and two ...
2
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1answer
57 views

Regarding the possibility of Closed Timelike Curves

I've been looking a lot at Closed Timelike Curves, and how if a theory allows for these curves it doesn't respect causality. I understand that about the curves themselves (Grandfather Paradox), but ...
2
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1answer
100 views

Hypersurface Normal

Could anyone explain why $$n^{a}n_{a}=\pm1$$ where $n^{a}$ is the normal to the hypersurface
8
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1answer
163 views

How to prove that a spacetime is maximally symmetric?

In Carroll's book on general relativity, I found the following remark: In two dimensions, finding that $R$ is a constant suffices to prove that the space is maximally symmetric [...] In higher ...
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1answer
120 views

How can a black hole have spin?

How is it possible, or even meaningful, to say that a black hole has spin? (Tangentially, if the singularity is assumed to be a point, it must have either zero or infinite angular momentum, in both ...
15
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3answers
262 views

Comparing predictions and reality for the gravitational attraction due to light beams

While doing some on-the-side reading, I stumbled across this question: Do two beams of light attract each other in general theory of relativity?. Great question and a great, easily understandable ...
2
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1answer
58 views

Energy-momentum tensor for dust

We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0v^\alpha v^\beta,$ where $\rho_0$ is the mass density in the dust's rest frame and $v^α$ is the dust's ...
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0answers
93 views

Positive Mass Theorem

I'm currently a third year undergrad writing about Minimal Surfaces. In particular, trapped surfaces and black holes. What does the Positive Mass Theorem have to do with this? And does the theorem ...
2
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2answers
1k views

How is Big Bang related to theory of relativity?

I'm not someone with good scientific knowledge, so if my question are weird, correct me. I was reading about big bang and I came by the theory of relativity. Can someone explain the relation between ...
11
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1answer
507 views

Is this a quaternion representation of the equations of motion of General Relativity?

In The Quaternion Group and Modern Physics by P.R. Girard, the quaternion form of the general relativistic equation of motion is derived from $du'/ds = (d a / d s ) u {a_c}^* + a u ( d {a_c}^* / ...
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1answer
60 views

Is conformal time observable?

The standard FRW metric with cosmic time is $$ ds^2 = -dt^2 + a^2(t)(\gamma_{ij}dx^i dx^j),$$ and we can measure $t$ as the proper time for comoving observers. Thus it makes sense to talk about the ...
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0answers
42 views

Information paradox and spacelike slices

I'm reading S. Mathur's paper on the information paradox and I can't seem to understand the reason why we choose spacelike slices. Is it because we want to have a global timelike vector so that we ...
0
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1answer
36 views

Why will choice of coordinates impose functional relations on the metric?

I am reading Steven Weinberg's Gravitation and Cosmology. On page 10 he says: In $D$ dimensions there will be $D(D+1)/2$ independent metric functions $g_{ij}$, and our freedom to choose the $D$ ...
12
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1answer
297 views

In there such a thing as the Black Hole Information Paradox?

When I first heard about the black hole information paradox, I thought it had no content. At the time, papers about it had been written for numerous years and they keep on coming. Now that the press ...
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1answer
52 views

Sobolev norm for Schwarzschild metric

Considering a static spacetime of the metric form \begin{equation} \mathrm{d}s^{2}=-V^{2}\mathrm{d}t^{2}+h_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j} \end{equation} with a timelike killing field ...
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1answer
86 views

Stress-energy tensor explicitly in terms of the metric tensor

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $R_{\mu\nu}$ and scalar curvature ...
2
votes
1answer
503 views

Equation for null geodesic around schwarzschild metric?

I'm trying to find the path of a photon around the Schwarzschild black hole, given its initial conditions. After much tribulation, I've basically given up on solving the equations by myself. I just ...
6
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1answer
93 views

Intuition for actions written as integrals over spacetime

Right now I'm simply looking for an intuitive explaination of actions that integrate over a 4-volume element, $d^4x$ rather than a parameter say $\lambda$. More specifically I'm well versed in action ...
4
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1answer
140 views

Effective mass of a black hole?

Suppose a black hole forms from a given mass of particles such as the core of a star going supernova. The black hole formed will have an effective mass due to the curvature of space time induced. Such ...
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0answers
33 views

Meaning of $k$ in Sachs-Wolfe formula for angular power spectrum

I've seen the formula for the angular power spectrum of the CMB written as $$C_\ell = \frac2\pi \int\left|\Theta_\ell(k) \right|^2 k^2dk, $$ where $\Theta_\ell(k)$ is the temperature contrast at a ...
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2answers
126 views

Unitary representations of the diffeomorphism group in curved spacetime

In (special) relativistic quantum mechanics there is a standard argument that says that the (rigged) Hilbert space of states $H$ should be equipped with a projective unitary representation $U$ of the ...
4
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1answer
86 views

Setting $\delta R =0$ on boundary of hypersurface

Does requiring $\delta R=0$ on the boundary of hyper-surface create any restrictions or problems in deriving the field equations from Einstein-Hilbert Action?